Question

Indicate which of the following is independent of the path by which a change occurs: (a) the change in potential energy when a book is transferred from table to shelf, (b) the heat evolved when a cube of sugar is oxidized to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g),(\mathbf{c})\) the work accomplished in burning a gallon of gasoline.

Short answer

The processes (a) change in potential energy when transferring a book from table to shelf and (b) heat evolved when a cube of sugar is oxidized are independent of the path by which the change occurs, as their outcomes depend only on the initial and final states. In contrast, (c) work accomplished in burning a gallon of gasoline is not independent of the path, as it depends on various factors such as engine efficiency and the combustion process.

Step-by-step solution

Process (a): Change in potential energy

The change in potential energy is calculated as the difference between the potential energy at the final position (shelf) and the potential energy at the initial position (table). This depends only on the height difference, the mass of the book, and the gravitational constant. No matter which path the book takes from the table to the shelf, this value will be the same. Therefore, the change in potential energy when transferring a book from table to shelf is independent of the path by which the change occurs.

Process (b): Heat evolved during sugar oxidation

The heat evolved during the oxidation of sugar is released due to the chemical reaction of sugar with oxygen and the formation of CO2 and H2O. The heat released in this reaction is determined by the changes in the chemical bonds within the molecules and does not depend on how the reaction occurs (i.e., the pathway through which sugar is oxidized). This means that the heat evolved during sugar oxidation is independent of the path by which the change occurs.

Process (c): Work accomplished in burning a gallon of gasoline

The work accomplished in burning a gallon of gasoline involves converting the chemical energy stored in gasoline into mechanical work. The amount of work done depends on various factors such as efficiency of the engine, the specific combustion process, and other external factors. This usually means that the amount of work accomplished can vary based on the path through which the gasoline is burned. As a result, the work accomplished in burning a gallon of gasoline is not independent of the path by which the change occurs. So, among the given processes, only (a) change in potential energy when transferring a book from table to shelf and (b) heat evolved when a cube of sugar is oxidized are independent of the path by which the change occurs.

Key concepts

Potential Energy

When we talk about potential energy, it refers to the energy stored in an object because of its position or state. Consider a book on the table. It has potential energy due to its elevation from the ground and gravity's influence.
The formula used to calculate potential energy is \( PE = mgh \), where:

• \( PE \) is the potential energy
• \( m \) is the mass of the object (in kilograms)
• \( g \) is the acceleration due to gravity (usually 9.8 m/s² on Earth)
• \( h \) is the height of the object above a reference point (in meters)

This way, potential energy becomes related only to the height, mass, and gravity. These factors depend on the start and end positions. It doesn't matter how you move between them. This is why a change in a book's potential energy going from the table to a shelf is path-independent. Simply put, it doesn't matter if you take a straight line or a curvy path. The potential energy relies solely on its initial and final positions.

Oxidation Reaction

An oxidation reaction is a chemical reaction in which a substance loses electrons, often in reaction with oxygen. Think of it like a trade where electrons are transferred. For instance, when sugar is oxidized, it combines with oxygen to produce carbon dioxide (\( \text{CO}_2 \)) and water (\( \text{H}_2 \text{O} \)).
During this reaction, the sugar's chemical bonds transform, releasing energy in the form of heat. The magic here lies in the fact that the energy released doesn't depend on how the reaction occurs but rather on the initial and final states of the molecules. So, regardless of how you oxidize that cube of sugar, the same amount of heat is released.
This is an essential principle in thermodynamics, known as a state function. Energy related to state functions, like heat in oxidation, only requires knowing where you started and ended, not how you got there.

Work-Energy Principle

The work-energy principle in thermodynamics tells us that the work done by forces on an object changes its energy. In other words, whenever you do work on something, you're transferring energy to or from it. This is especially relevant when looking at engines or systems involving combustion, like burning gasoline.
The relation between work and energy can be expressed as:
\[ W = \Delta KE + \Delta PE \]

• \( W \) is the work done by the system
• \( \Delta KE \) is the change in kinetic energy
• \( \Delta PE \) is the change in potential energy

When you burn fuel, like gasoline, chemical energy transforms into mechanical work. This transformation means turning chemical potential energy into usable work. However, this work depends on the combustion pathway, engine efficiency, and external conditions. Unlike potential energy changes, work is path-dependent because it considers the specific way energy changes occur and the system's efficiency.
Thus, burning gasoline doesn't have a fixed value of work done, as it may vary with different engines or conditions.